An. Şt. Univ. Ovidius Constanţa
Vol. 20(3), 2012, 95–110
Equilibrium existence under generalized
convexity
Monica Patriche
Abstract
We introduce, in the first part, the notion of weakly convex pair of
correspondences, we give its economic interpretation, we state a fixed
point and a selection theorem. Then, by using a tehnique based on a
continuous selection, we prove existence theorems of quilibrium for an
abstract economy. In the second part, we define the weakly biconvex
correspondences, we prove a selection theorem and we also demonstrate
the existence of equilibrium for a generalized quasi-game (2003 Kim’s
model). In the last part of the paper, we give other applications in the
game theory, finding equilibrium for abstract economies having correspondences with weakly convex graph. We show that the equilibrium
exists without continuity assumptions.
1
Introduction
An open problem in the equilibrium theory is to prove the existence of fixed
points for correspondences under nonconvexity (in the usual sense) assumptions. Some results on this subject were obtained by C. D. Horvath [7], G.
Tian [13], X. Ding, He Yiran [3] or K. Wlodarczyk and D. Klim [15], [16].
The aim of this paper is to prove a fixed point and a selection theorem under
generalized covexity conditions and to give an application in the game theory.
The importance of these results also consists of the fact that the existence of
Key Words: weakly convex pairs of correspondence, weakly biconvex correspondences,
weakly convex graph, fixed point theorem, continuous selection, abstract economy, equilibrium.
2010 Mathematics Subject Classification: 91B52, 91B50, 91A80.
Received: April, 2011.
Revised: April, 2011.
Accepted: February, 2012.
95
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fixed points and of the equilibrium takes place without continuity properties
of the involved correspondences.
Within the last years, many authors generalized the classical models of
abstract economy due to A. Borglin and H. Keiding [2], W. Shafer and H.
Sonnenschein [12] or N. C. Yannelis and N. D. Prahbakar [18]. Also, W.K.
Kim [8] obtained a generalization of the quasi fixed-point theorem due to I.
Lefebvre [9]and, as an application, he proved an existence theorem of equilibrium for a generalized quasi-game with infinite number of agents. W.K.Kim’s
result concerns generalized quasi-games where the strategy sets are metrizable
subsets in linear topological convex spaces.
In the first part of this paper we introduce the notion of weakly convex pair
of correspondences, we give its economic interpretation, we state a fixed point
and a selection theorem. Then, by using a tehnique based on a continuous
selection, we prove existence theorems of quilibrium for an abstract economy.
In the second part, we define the weakly biconvex correspondences, we prove
a selection theorem and we also demonstrate the existence of equilibrium for
a generalized quasi-game (2003 Kim’s model). In the last part of the paper
we give other applications in the game theory, finding equilibrium for abstract
economies having correspondences with weakly convex graph. We show that
the equilibrium exists without continuity assumptions.
Bi-convexity was studied by R. Aumann, S. Hart in [1] or J. Gorski, F.
Pfeuffer, K. Klamroth in [5]. We continue our work on studying the existence
conditions of equilibrium of quasi-games [10] or the existence of fixed points
for correspondences [11].
The paper is organized in the following way: Section 2 contains preliminaries and notation. The weakly convex pairs of correspondences are studied
in Section 3. Biconvexity of the correspondences, W. K. Kim’s model of quasigame and the quasi-equilibrium existence results are presented in Section 4.
The equilibrium theorems for correspondences with the weakly convex graph
selection property are stated in Section 5.
2
Preliminaries and notation
Let A be a subset of a topological space X. 2A denotes the family of all subsets
of A. cl A denotes the closure of A in X. If A is a subset of a vector space,
coA denotes the convex hull of A. If F , T : A → 2X are correspondences, then
coT , cl T , T ∩ F : A → 2X are correspondences defined by (coT )(x) =coT (x),
(clT )(x) =clT (x) and (T ∩ F )(x) = T (x) ∩ F (x) for each x ∈ A, respectively.
The graph of T : X → 2Y is the set Gr(T ) = {(x, y) ∈ X × Y | y ∈ T (x)}.
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Equilibrium existence under generalized convexity
The correspondence T is defined by T (x) = {y ∈ Y : (x, y) ∈clX×Y GrT }
(the set clX×Y Gr(T ) is called the adherence of the graph of T ). It is easy to
see that clT (x) ⊂ T (x) for each x ∈ X.
Lemma 1. [19] Let X be a topological space, Y be a non-empty subset of
a topological vector space E, ß be a base of the neighborhoods of 0 in E and
A : X → 2Y . For each V ∈ß, let AV : X → 2Y be defined by AV (x) = (A(x) +
V ) ∩ Y for each x ∈ X. If x
b ∈ X and yb ∈ Y are such that yb ∈ ∩V ∈ß AV (b
x),
then yb ∈ A(b
x).
Definition 1. Let X, Y be topological spaces and T : X → 2Y be a correspondence
1. T is said to be upper semicontinuous if for each x ∈ X and each open
set V in Y with T (x) ⊂ V , there exists an open neighborhood U of x in X
such that T (y) ⊂ V for each y ∈ U .
2. T is said to be lower semicontinuous if for each x∈ X and each open
set V in Y with T (x) ∩ V ̸= ∅, there exists an open neighborhood U of x in
X such that T (y) ∩ V ̸= ∅ for each y ∈ U .
Lemma 2. [14] Let X be a topological space, Y be a topological linear space,
and let A : X → 2Y be an upper semicontinuous correspondence with compact
values. Assume that the sets C ⊂ Y and K ⊂ Y are closed and respectively
compact. Then T : X → 2Y defined by T (x) = (A(x) + C) ∩ K for all x ∈ X
is upper semicontinuous.
To prove our theorems, we need Wu’s theorem:
Theorem 3. [17] Let I be an index set. For each i ∈ I, let Xi be a nonempty
convex subset of a Hausdorff locally convex topological vector∏space Ei , Di a
Xi → 2Di two
non-empty compact metrizable subset of Xi and Si , Ti : X :=
i∈I
correspondences with the following conditions:
(i) for each x ∈ X, coSi (x) ⊂ Ti (x) and Si (x) ̸= ∅,
(ii) Si is lower semicontinuous.
∏
∏
Then, there exists a point x =
xi ∈ D =
Di such that xi ∈ Ti (x) for
each i ∈ I.
i∈I
i∈I
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Weakly convex pairs of correspondences
Notation. Let ∆n−1 = {(λ1 , λ2 , ..., λn ) ∈ Rn :
n
∑
λi = 1 and λi > 0, i =
i=1
1, 2, ...,
n} be the standard (n-1)-dimensional simplex in Rn .
We introduce the following notion.
Definition 2. Let X be a convex set in a topologiacl vector space E, Y be
a nonempty subset of a topological vector F and S, T : X → 2Y two correspondences. (S, T ) is called weakly convex pair of correspondences if, for each
finite set {x1 , x2 , ..., xn } ⊂ X, there exists yi ∈ S(xi ) , (i = 1, 2, ..., n) such
n
n
∑
∑
that for every λ1 , λ2 , ..., λn ∈ ∆n−1 , then y =
λi yi ∈ T ( λi xi ).
i=1
3.1
i=1
A fixed point theorem
We state the following fixed point theorem:
Theorem 4. Let Y be a non-empty subset of a topological vector space E and
K be a (n − 1)- dimensional simplex in E. Let (S, T ) : K → 2Y be a weakly
convex pair of correspondences and s : Y → K be a continuous function.
Then, there exists x∗ ∈ K such that x∗ ∈ s ◦ T (x∗ ).
Proof. Let a1 , a2 , ..., an be the vertices of K. Since (S, T ) is weakly convex
pair of correspondences, there exist bi ∈ S(ai ), such that for every (λ1 , λ2 , ..., λn ) ∈
n
n
∑
∑
∆n−1 , then y =
λi bi ∈ T ( λi ai ).
i=1
i=1
Since K is a (n − 1)-dimensional simplex with the vertices a1 , ..., an , there
exists unique continuous functions λi : K → R, i = 1, 2, ..., n such that for
n
∑
each x ∈ K, we have (λ1 (x), λ2 (x), ..., λn (x)) ∈ ∆n−1 and x =
λi (x)ai .
Let’s define f : K → 2Y by
f (ai ) = bi (i = 1, ..., n) and
n
n
n
∑
∑
∑
f ( λi ai ) =
λi bi ∈ T ( λi ai ).
i=1
i=1
i=1
i=1
We show that f is continuous.
Let (xm )m∈N be a sequence which converges to x0 ∈ K, where xm =
n
n
∑
∑
λi (xm )ai and x0 =
λi (x0 )ai . By the continuity of λi , it follows that for
i=1
i=1
each i = 1, 2, ..., n, λi (xm ) → λi (x0 ) as m → ∞. Hence f (xm ) → f (x0 ) as
m → ∞, i.e. f is continuous.
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Equilibrium existence under generalized convexity
Since s : Y → K is continuous, we obtain that s◦f : K → K is continuous.
According to Brouwer’s fixed point theorem, there exists a point x∗ ∈ K such
that x∗ = s ◦ f (x∗ ) and then, x∗ ∈ s ◦ T (x∗ ).
Theorem 5. (selection theorem). Let Y be a non-empty subset of a topological
vector space E and K be a (n − 1)- dimensional simplex in a topological vector
space F. Let (S, T ) : K → 2Y be a weakly convex pair of correspondences.
Then, T has a continuous selection on K.
3.2
Economic interpretation
We consider an abstract economy with I - the set of agents. Each agent can
choose
a strategy from a set Xi and has a preferrence correspondence
Pi : X =
∏
∏
Xi → Xi and a constraint correspondence Ai : X =
Xi → 2Xi . The
i∈I
i∈I
traditional approach considers that the preferrence of agent i is characterized
by a binary relation ≽i on the set Xi . A real valued function ui that satisfies
x ≽i y ⇔ ui (x) ≥ ui (y) is called an utility function of the preferrence ≽i . The
relation between the utility function ui and the preferrence correspondence for
each agent i is:
Pi (x) = {yi ∈ Ai (x) : ui (x, yi ) > ui (x, xi )} , where, in this case, ui : X ×
Xi → Xi .
The aim of the equilibrium theory is to maximize each agent’s utility on
a convex strategy set. For that, the notion of convexity of the preferrence is
very important:
Definition 3. The preferrence ≽ is called convex if x ≽ y implies λx + (1 −
λ)y ≽ y for λ ∈ [0, 1].
The intuitive interpretation is that, given two strategies x and y, the composed strategy λx + (1 − λ)y ≽ y with λ ∈ [0, 1] is more valuable if x is
already preferrable to y. For an abstract economy, if we have yi ∈ Ai (x) and
ui (x, yi ) > ui (x, xi ), if the preferrence ≽i (and then the utility function ui )
is convex, we obtain that
if yi ∈ Ai (x) then ui (x, λyi + (1 − λ)xi ) > ui (x, xi )
or, equivalently,
if yi ∈ Pi (x) then λyi + (1 − λ)xi ∈ Pi (x) if we have that λyi + (1 − λ)xi ∈
Ai (x).
For the case that, for the index i, (Ai , Pi ) is a weakly convex pair of
correspondences, the interpretation is the following: for every x1 , x2 , ..., xn ∈
X, there exist yi1 ∈ Ai (x1 ), yi2 ∈ Ai (x2 ), ..., yin ∈ Ai (xn ), such that, for each λ ∈
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∆n−1 , there exists yi =
n
∑
λk yik with the property that yi ∈ Pi (
k=1
if there exists the utility function ui : yi ∈ Ai (
ui (
n
∑
λk xk , (
k=1
n
∑
n
∑
λk xk ) and ui (
k=1
n
∑
λk xk ) (i.e.,
k=1
n
∑
λk xk , yi ) >
k=1
λk xk )i ).
k=1
We introduce the notion of weakly convex preferrence.
Definition 4. The preferrence ≽ is called weakly convex if for each y ∈ X,
there exists x ∈ X such that for each λ ∈ [0, 1] we have that λx + (1 − λ)y ≽ y.
3.3
Applications in the equilibrium theory
First, we present the model of an abstract economy and the definition of an
equilibrium.
Let I be a non-empty set (the set of agents). For each i ∈ I, let Xi be
a non-empty ∏
topological vector space representing the set of actions and let’s
Xi ; let Ai , Bi : X → 2Xi be the constraint correspondences
define X :=
i∈I
and Pi the preference correspondence.
Definition 5. The family Γ = (Xi , Ai , Pi , Bi )i∈I is said to be an abstract
economy.
Definition 6. An equilibrium for Γ is defined as a point x∗ ∈ X such that for
each i ∈ I, x∗i ∈ B i (x∗ ) and Ai (x∗ , ) ∩ Pi (x∗ ) = ∅.
Remark 1. When for each i ∈ I, Ai (x) = Bi (x) for all x ∈ X, this abstract economy model coincides with the classical one introduced by Borglin
and Keiding in [2]. If in addition, B i (x∗ ) =clXi Bi (x∗ ) for each x ∈ X, which
is the case if Bi has a closed graph in X × Xi , the definition of an equilibrium
coincides with that one used by Yannelis and Prabhakar [18].
For the following theorems, we will use the selection theorem and a tehnique
based on a continuous selection. We show the existence of equilibrium for an
abstract economy without assuming the continuity of the constraint and of
the preference correspondences Ai and Pi .
First, we prove a new equilibrium existence theorem for a noncompact
abstract economy with constraint and preference correspondences Ai and Pi ,
which have the property that their intersection Ai ∩Pi contains a selector Si on
the domain Wi of Ai ∩ Pi , (Ai , Si ) is a weakly convex pair of correspondences
and Wi must be a simplex. To find the equilibrium point, we use Wu’s fixed
point theorem [17].
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Equilibrium existence under generalized convexity
Theorem 6. Let Γ = (Xi , Ai , Pi , Bi )i∈I be an abstract economy, where I is
a (possibly uncountable) set of agents such that for each i ∈ I :
(1) Xi is a non-empty convex set in a locally convex space Ei and there exists a compact subset Di of Xi∏
containing all the values of the correspondences
Ai , Pi and Bi such that D =
Di is metrizable;
i∈I
(2) clBi is lower semicontinuous, has non-empty convex values and for
each x ∈ X, Ai (x) ⊂ Bi (x);
(3) Wi = {x ∈ X / (Ai ∩ Pi ) (x) ̸= ∅} is a (ni − 1)-dimensional simplex
in X such that Wi ⊂coD;
(4) there exists a correspondence Si : Wi → 2Di such that Si (x) ⊂ (Ai ∩ Pi ) (x)
for each x ∈ Wi and (Ai , Si ) is a weakly convex pair of correspondences;
(5) for each x ∈ Wi , xi ∈
/ (Ai ∩ Pi )(x).
Then there exists an equilibrium point x∗ ∈ D for Γ, i.e., for each i ∈ I,
x∗i ∈clBi (x∗ ) and Ai (x∗ ) ∩ Pi (x∗ ) = ∅.
Proof. Let be i ∈ I. From the assumption (4) and the selection theorem 3,
it follows that there exists a continuous function fi : Wi → Di such that for
each x ∈ Wi , fi (x) ∈ Si (x) ⊂ Ai (x) ∩ Pi (x) ⊂ Bi (x).
{
{fi (x)}, if x ∈ Wi ,
Di
Let’s define the correspondence Ti : X → 2 , by Ti (x) :=
clBi (x), if x ∈
/ Wi ;
Ti is lower semicontinuous on X.
Let V be an closed subset of Xi , then
U := {x ∈ X | Ti (x) ⊂ V } ={x ∈ Wi | Ti (x) ⊂ V } ∪ {x ∈ X \ Wi |
Ti (x) ⊂ V }
={x ∈ Wi | fi (x) ∈ V } ∪ {x ∈ X | clBi (x) ⊂ V }
=(fi−1 (V ) ∩ Wi ) ∪ {x ∈ X | clBi (x) ⊂ V } .
U is a closed set, because Wi is closed, fi is a continuous map on int
∏X Ki
Di .
and the set {x ∈ X | clBi (x) ⊂ V } is closed since clBi is l.s.c. Let D =
i∈I
Then, according to Tychonoff’s Theorem, D is compact in the convex set X.
By Wu’s fixed-point theorem in [17], applied for the correspondences Si =
Ti and Ti : X → 2Di , there exists x∗ ∈ D such that for each i ∈ I, x∗i ∈ Ti (x∗ ).
If x∗ ∈ Wi for some i ∈ I, then x∗i = fi (x∗ ), which is a contradiction.
Therefore, x∗ ∈
/ Wi , and hence (Ai ∩ Pi )(x∗ ) = ∅. Also, for each i ∈ I, we
∗
∗
have xi ∈ Ti (x ), and then x∗i ∈clBi (x∗ ).
For Theorem 5, we use an approximation method, in the meaning that
we obtain, for each i ∈ I, a continuous selection fiVi of (Ai +∏Vi ) ∩ Pi , where
Vi is a convex neighborhood of 0 in Xi . For every V =
Vi , we obtain
i∈I
an equilibrium point for the associated approximate abstract economy ΓV =
(Xi , Ai , Pi , BVi )i∈I , i.e., a point x∗ ∈ X such that Ai (x∗ ) ∩ Pi (x∗ ) = ∅
and x∗i ∈ BVi (x∗ ), where the correspondence BVi : X → 2Xi is defined by
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BVi (x) =cl(Bi (x) + Vi ) ∩ Xi for each x ∈ X and for each i ∈ I. Finally, we use
Lemma 1 to get an equilibrium point for Γ in X. The compactness assumption
for Xi is essential in the proof.
Theorem 7. Let Γ = (Xi , Ai , Pi , Bi )i∈I be an abstract economy, where I is
a (possibly uncountable) set of agents such that for each i ∈ I :
(1) Xi is a non-empty compact convex set in a locally convex space Ei ;
(2) clBi is upper semicontinuous, has non-empty convex values and for
each x ∈ X, Ai (x) ⊂ Bi (x);
(3) the set Wi : = {x ∈ X / (Ai ∩ Pi ) (x) ̸= ∅} is non-empty, open and
Ki =clWi is a (ni − 1)-dimensional simplex in X ;
(4) For each convex neighbourhood V of 0 in Xi , (Ai , (Ai + V ) ∩ Pi ) is a
weakly convex pair of correspondences, where (Ai + V ) ∩ Pi : Ki → 2Xi ;
(5) for each x ∈ Ki , xi ∈
/ Pi (x).
Then there exists an equilibrium point x∗ ∈ X for Γ, i.e., for each i ∈ I,
x∗i ∈ B i (x∗ ) and Ai (x∗ ) ∩ Pi (x∗ ) = ∅.
Proof. For each i ∈ I, let ßi denote the
∏family of all open convex neighßi . Since (Ai , (Ai + V ) ∩ Pi ) is a
borhoods of zero in Ei . Let V = (Vi )i∈I ∈
i∈I
weakly convex pair of correspondences on Ki , then, from the selection theorem
3, there exists a continuous function fiVi : Ki → Xi such that for each x ∈ Ki ,
fiVi (x) ∈ (Ai (x) + Vi ) ∩ Pi (x) ⊂ (Ai (x) + Vi ) ∩ Xi .
It follows that fiVi (x) ∈cl(Bi (x) + Vi ) for x ∈ Ki . Since Xi is compact, we
have that cl(Bi (x)) is compact for every x ∈ X and cl(Bi (x)+Vi ) =cl(Bi (x))+clVi
for every Vi ⊂ Ei .
Let’s define the correspondence TiVi : X → 2Xi , by
{
{fiVi (x)},
if x ∈ intX K = Wi ,
Vi
Ti (x) :=
cl(Bi (x) + Vi ) ∩ Xi , if x ∈ X r intX Ki ;
The correspondence BVi : X → 2Xi , defined by BVi (x) :=cl(Bi (x)+Vi )∩Xi
is u.s.c. by Theorem 1.1 in [14]. Then following the same line as in Theorem
4, we can prove that TiVi is upper semicontinuous on X and has closed convex
values.
∏ Vi
Ti (x) for each x ∈ X.
Let’s define T V : X → 2X by T V (x) :=
i∈I
T V is an upper semicontinuous correspondence and also has non-empty
convex closed values.
Since X is a compact convex set, according to Fan’s fixed-point Theorem
[4], there exists x∗V ∈ X such that x∗V ∈ T V (x∗V ), i.e., for each i ∈ I, (x∗V )i ∈
TiVi (x∗V ).
∪
We state that x∗V ∈ X \ intX Ki .
i∈I
Equilibrium existence under generalized convexity
103
If x∗V ∈intX Ki , (x∗V )i ∈ TiVi (x∗V ) = fi (x∗V ) ∈ ((Ai (x∗V ) + Vi ) ∩ Pi )(x∗V ) ⊂
Pi (x∗V ), which contradicts assumption (5).
Hence (x∗V )i ∈cl(Bi (x∗V ) + Vi ) ∩ Xi and (Ai ∩ Pi )(x∗V ) = ∅, i.e. x∗V ∈ QV
where
QV = ∩i∈I {x ∈ X : xi ∈cl(Bi (x) + Vi ) ∩ Xi and (Ai ∩ Pi )(x) = ∅}.
Since Wi is open, QV is the intersection of non-empty closed sets, then it
is non-empty, closed in X.
∏
We prove that the family {QV : V ∈
ßi } has the finite intersection
i∈I
property.
∏
(k)
Let {V (1) , V (2) , ...V (n) } be any finite set of
ßi and let V (k) = (Vi )i∈I ,
i∈I
n
(k)
k = 1, ...n. For each i ∈ I, let Vi = ∩ Vi , then Vi ∈ ßi ; thus V = (Vi )i∈I ∈
k=1
n
n
∏
ßi . Clearly QV ⊂ ∩ QV (k) so that ∩ QV (k) ̸= ∅.
k=1
k=1
i∈I
∏
Proof. Since X is compact and the family {QV : V ∈
ßi } has the finite
i∈I
∏
ßi } ̸= ∅. Let’s take any
intersection property, we have that ∩{QV : V ∈
i∈I
∏
x∗ ∈ ∩{QV : V ∈ ßi }, then for each i ∈ I and each Vi ∈ ßi , x∗i ∈cl(Bi (x∗ ) +
i∈I
Vi ) ∩ Xi and (Ai ∩ Pi )(x∗ ) = ∅; but then x∗ ∈cl(Bi (x∗ )) according to Lemma
1 and (Ai ∩ Pi )(x∗ ) = ∅ for each i ∈ I so that x∗ is an equilibrium point of
Γ in X. In the theorem above, the correspondences Ai ∩ Pi don’t verify continuity
assumptions and do not have convex or compact values. The importance of
our results also consists in the fact that the existence of fixed points and of the
equilibrium takes place without continuity properties of the correspondences
involved.
4
4.1
Biconvexity of the correspondences and applications
in the game theory
Preliminaries
Let X ⊂ E1 and Y ⊂ E2 be two nonempty, convex sets, E1 , E2 are topological
vector space and let B ⊂ X × Y. The y− and x− sections of B are defined as
follows:
Bx := {y ∈ Y : (x, y) ∈ B}
By := {x ∈ X : (x, y) ∈ B}
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Definition 7. The set B ⊂ X × Y is called a biconvex set on X × Y if Bx is
convex for every x ∈ X and By is convex for every y ∈ Y.
Definition 8. Let (xi , yi ) ∈ X × Y for i = 1, 2, ...n. A convex combination
n
n
∑
∑
(x, y) =
λi (xi , yi ), (with
λi = 1, λi ≥ 0 i = 1, 2, ..., n) is called biconvex
i=1
i=1
combination if x1 = x2 = ... = xn = x or y1 = y2 = ... = yn = y.
The following characterization for biconvex sets was formulated by Aumann and Hart:
Theorem 8. [1] A set B ⊆ X × Y is biconvex if and only if B contains all
biconvex combinations of its elements.
As in the convex case, it is possible to define the biconvex hull of a given
set A ⊆ X × Y .
∩
Definition 9. Let A ⊆ X × Y be a given set. The set H := { AI : A ⊆ AI ,
AI is biconvex} is called the biconvex hull of A and is denoted biconv(A).
Aumann and Hart stated the following properties of the set H:
Theorem 9. [1] The above defined set is biconvex. Furthermore, H is the
smallest biconvex set (in the sense of set inclusion), which contains A.
As biconvex combinations are, by definition, a special case of convex combinations and the convex hull conv(A) of a given set A consists of all convex
combinations of the elements of A, we have:
Lemma 10. Let A ⊆ X × Y be a given set. Then biconv(A) ⊆conv(A).
Aumann and Hart proposed an inductively way to construct the biconvex
hull of a given set A. They defined the sequence {An }n∈N as follows:
A1 := A;
An+1 := {(x; y) ∈ An : (x, y) is a biconvex combination of elements of An }.
′
Let H := ∪n∈N An denote the limit of this sequence.
Proposition 11. [1] The above constructed set H ′ is biconvex and equals H,
the biconvex hull of A.
We introduce the following definition.
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Equilibrium existence under generalized convexity
Definition 10. Let B ⊂ X × Y be a biconvex set, Z a nonempty subset of
a topological vector space F and T : B → 2Z a correspondence. T is called
weakly biconvex if for each finite set {(x1 , y1 ), (x2 , y2 ), ..., (xn , yn )} ⊂ B, there
exists zi ∈ T (xi , yi ) , (i = 1, 2, ..., n) such that for every biconvex combination
n
n
∑
∑
(x, y) =
λi (xi , yi ) ∈ B (with
λi = 1, λi ≥ 0 i = 1, 2, ..., n), then
y=
n
∑
i=1
λi zi ∈ T (
i=1
n
∑
i=1
λi (xi , yi )).
i=1
We formulate the following fixed point theorem for weakly biconvex correspondences.
Theorem 12. Let Y be a non-empty subset of a topological vector space F and
K ⊂ E1 × E2 , where E1 , E2 are topological vector spaces. Suppose that K is
the biconvex hull of {(a1 , b1 ), (a2 , b2 ), ..., (an , bn )} ⊂ E1 × E2 . Let T : K → 2Y
be a weakly biconvex correspondence and s : Y → K be a continuous function.
Then, there exists x∗ ∈ K such that x∗ ∈ s ◦ T (x∗ ).
Proof. Since T is weakly biconvex, there exist ci ∈ T (ai , bi ), (i = 1, 2, ..., n),
n
∑
such that, for every (λ1 , λ2 , ..., λn ) ∈ ∆n−1 , there exists z ∈ T ( λi (ai , bi ))
with z =
n
∑
i=1
λi zi .
i=1
Since K is the biconvex hull of (a1 , b1 ), ..., (an , bn ), there exists unique
continuous functions λi : K → R, i = 1, 2, ..., n such that for each (x, y) ∈ K,
n
∑
we have (λ1 (x, y), λ2 (x, y), ..., λn (x, y)) ∈ ∆n−1 and (x, y) =
λi (x, y)(ai , bi ).
i=1
Let’s define f : K → 2Y by
f (ai , bi ) = ci (i = 1, ..., n) and
n
n
∑
∑
f ( λi (ai , bi )) =
λi ci ∈ T (x, y).
i=1
n
∑
i=1
We show that f is continuous.
Let (xm , ym )m∈N be a sequence which converges to x0 ∈ K, where (xm , ym ) =
λi (xm , ym )(ai , bi ) implies a1 = a2 = ... = an = a or b1 = b2 = ... = bn = b
i=1
and (x0 , y0 ) =
n
∑
λi (x0 )(ai , bi ) with a1 = a2 = ... = an = a or b1 = b2 =
i=1
... = bn = b. By the continuity of λi , it follows that for each i = 1, 2, ..., n,
λi (xm , ym ) → λi (x0 .y0 ) as m → ∞. Hence f (xm , ym ) → f (x0 , y0 ) as m → ∞,
i.e. f is continuous.
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Monica Patriche
Since s : Y → K is continuous, we obtain that s◦f : K → K is continuous.
According to Brouwer’s fixed point theorem, there exists a point x∗ ∈ K such
that x∗ = s ◦ f (x∗ ) and then, x∗ ∈ s ◦ T (x∗ ).
Theorem 13. (selection theorem). Let Y be a non-empty subset of a topological vector space F and K ⊂ E1 ×E2 , where E1 , E2 are topological vector spaces.
Suppose that K is the biconvex hull of {(a1 , b1 ), (a2 , b2 ), ..., (an , bn )} ⊂ E1 ×E2 .
Let T : K → 2Y be a weakly biconvex correspondence. Then, T has a continuous selection on K.
In order to prove the existence of equilibrium, we need the following theorem:
Theorem 14. ([10]). Let I and J be any (possibly uncountable) index sets.
For each i ∈ I and j ∈ J, let Xi and Yj be non-empty compact convex subsets
of Hausdorff locally convex spaces Ei and respectivelly Fj .
∏
∏
Let X := Xi , Y :=
Yj and Z := X × Y .
i∈I
For each i ∈ I let Si : Z → 2Xi be a correspondence such that the set
Wi = {(x, y) ∈ Z | Si (x, y) ̸= ∅} is open and Si has a continuous selection fi
on Wi .
For each j ∈ J let Tj : Z → 2Yj be an upper semicontinuous correspondence with non-empty closed convex values.
Then there exists a point (x∗ , y ∗ ) ∈ Z such that for each i ∈ I, either
Si (x∗ , y ∗ ) = ∅ or x∗i ∈ Si (x∗ , y ∗ ), and for each j ∈ J, yj∗ ∈ Tj (x∗ , y ∗ ).
As a consequence, we have the following:
Corollary 15. Let I and J be any (possibly uncountable) index sets. For
each i ∈ I and j ∈ J, let Xi and Yj be non-empty compact convex subsets of
Hausdorff locally convex spaces Ei and respectivelly Fj .
∏
∏
Yj and Z := X × Y .
Let X := Xi , Y :=
i∈I
For each i ∈ I let Si : Z → 2Xi be a correspondence such that the
set Wi = {(x, y) ∈ Z | Si (x, y) ̸= ∅} is the interior of the biconvex hull of
{(a1 , b1 ), (a2 , b2 ), ...,
(an , bn )} ⊂ Z and Si is weakly biconvex on Wi .
For each j ∈ J let Tj : Z → 2Yj be an upper semicontinuous correspondence with non-empty closed convex values.
Then there exists a point (x∗ , y ∗ ) ∈ Z such that for each i ∈ I, either
Si (x∗ , y ∗ ) = ∅ or x∗i ∈ Si (x∗ , y ∗ ), and for each j ∈ J, yj∗ ∈ Tj (x∗ , y ∗ ).
Equilibrium existence under generalized convexity
4.2
107
Kim’s model of the generalized quasi-game and equilibrium
theorems
In this section, we study the following model of a generalized quasi-game.
Let I be a nonempty set (the set of agents). For each i ∈ I, let Xi be a
non-empty topological
vector space representing the set of actions and let’s
∏
define X :=
Xi ; let Ai , Bi : X ×X → 2Xi be the constraint correspondences
i∈I
and Pi : X × X → 2Xi the preference correspondence.
Definition 11. [10]. A generalized quasi-game Γ = (Xi , Ai , Bi , Pi )i∈I is defined as a family of ordered quadruples (Xi , Ai , Bi , Pi ).
In particular, when I={1, 2...n}, Γ is called the n-person quasi-game.
Definition 12. [10]. An equilibrium for Γ is defined as a point (x∗ , y ∗ ) ∈ X ×
X such that, for each i ∈ I, yi∗ ∈clBi (x∗ , y ∗ ) and Ai (x∗ , y ∗ ) ∩ Pi (x∗ , y ∗ ) = ∅.
If Ai (x, y) = Bi (x, y) for each (x, y) ∈ X × X and i ∈ I, this model
coincides with the one introduced by W. K. Kim [8].
If, in addition, for each i ∈ I, Ai , Pi are constant with respect to the first
argument, this model coincides with the classical one of the abstract economy
and the definition of equilibrium is that given in [18].
Now, we state the following equilibrium theorem for generalized quasigames with correspondences which does not have continuity properties.
Theorem 16. Let Γ = (Xi , Ai , Bi , Pi )i∈I be a generalized quasi-game where
I is a (possibly uncountable) set of agents such that for each i ∈ I :
(1) Xi is a non-empty ∏
compact convex set in a Hausdorff locally convex
Xi and Z := X × X;
space Ei and denote X :=
i∈I
(2) The correspondence Bi : Z → 2Xi is non-empty, convex valued such
that for each (x, y) ∈ Z, Ai (x, y) ⊂ Bi (x, y) and clBi is upper semicontinuous;
(3) (Ai , Ai ∩ Pi ) is a weakly biconvex pair of correspondences on Wi ;
(4) the set Wi : = {(x, y) ∈ Z / (Ai ∩ Pi ) (x, y) ̸= ∅} is the interior of the
biconvex hull of {(a1 , b1 ), (a2 , b2 ), ..., (an , bn )} ⊂ Z;
(5) for each (x, y) ∈ Wi , xi ∈coP
/
i (x, y).
Then there exists an equilibrium point (x∗ , y ∗ ) ∈ Z for Γ, i.e., for each
i ∈ I, yi∗ ∈clBi (x∗ , y ∗ ) and Ai (x∗ , y ∗ ) ∩ Pi (x∗ , y ∗ ) = ∅.
Proof. For each i ∈ I, we define Φi : Z → 2Xi by
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Monica Patriche
{
co(Ai ∩ Pi )(x, y), if (x, y) ∈ Wi ,
∅,
if (x, y) ∈
/ Wi ;
The restriction Ai ∩Pi/Wi : Wi → 2Xi is a weakly biconvex correspondence.
Then, applying Theorem 9, we can obtain that there exists a continuous selection fi : Wi → Xi such that fi (x, y) ∈ (Ai ∩ Pi )(x, y) for each (x, y) ∈ Wi .
For each j ∈ I, we define Ψj : Z → 2Xi , by Ψj (x, y) =clBj (x, y) for each
(x, y) ∈ Z.
Then Ψj is an upper semicontinuous correspondence and Ψj (x, y) is a nonempty, convex, closed subset of Xj for each (x, y) ∈ Z.
According to Theorem 10, it follows that there exists (x∗ , y ∗ ) ∈ Z such
that for each i ∈ I, either Φi (x∗ , y ∗ ) = ∅ or x∗i ∈ Φi (x∗ , y ∗ ) and for each
j ∈ J, yj∗ ∈ Ψj (x∗ , y ∗ ).
If x∗i ∈ Φi (x∗ , y ∗ ) for some i ∈ I, then x∗i ∈ Φi (x∗ , y ∗ ) = co(Ai ∩
Pi )(x∗ , y ∗ ) ⊂coPi (x∗ , y ∗ ) which contradicts the assumption (5).
Therefore, for each i ∈ I, Φi (x, y) = ∅ and then (x∗ , y ∗ ) ∈
/ Wi . Hence,
(Ai ∩ Pi )(x∗ , y ∗ ) = ∅ and for each i ∈ I, y ∗ ∈ Ψi (x∗ , y ∗ ) =clBi (x∗ , y ∗ ). Acknowledgment: This work was supported by the strategic grant POSDRU/89/1.5/S/58852, Project ”Postdoctoral programme for training scientific
researchers” cofinanced by the European Social Found within the Sectorial
Operational Program Human Resources Development 2007-2013.
Φi (x, y) =
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110
Monica PATRICHE,
Department of Mathematics,
University of Bucharest,
14 Academiei Street, 010014 Bucharest, Romania.
Email: monica.patriche@yahoo.com
Monica Patriche